Combine butter and brown sugar in a large bowl; beat with an electric mixer until light and fluffy. Add egg yolks one at a time, beating well after each addition. Beat in vanilla extract and salt.

Sprinkle xanthan gum over flour in a separate bowl; mix together very well. Beat flour mixture into the butter mixture until dough comes together. Cover with plastic wrap and refrigerate for 30 minutes.

Preheat the oven to 350 degrees F (175 degrees C). Grease 2 cookie sheets.

Mix coconut flakes and pecans in a shallow bowl. Beat egg whites in a small bowl.

Pinch off small pieces of dough and roll into balls. Dip into the egg whites then roll in the coconut mixture. Place 2 inches apart on the greased cookie sheets. Press 3 milk chocolate pieces into the center of each cookie, like eggs in a nest.

Bake in the preheated oven until edges are golden, 8 to 10 minutes. Transfer to a wire rack to cool.